Large differences in execution time between MATLAB and R

Question

I'm trying to implement a really simple 4^th order Runge-Kutta Method, for solving the ODE y'=f(x,y).

I've implemented the algorithm in both R and MATLAB (see below), but for some reason it takes a few minutes to run in MATLAB and a few milliseconds to run in R.

My question is WHY?

It appears that the only difference is the initialization, and playing around with that doesn't seem to make a difference.

---

R Script:

# Initialise Variables ----------------------------------------------------

L  = 1 #Domain of solution function
h  = 0.01#step size
x0 = 0
y0 = 0
x  = x0
y  = y0

# Define Forcing Function -------------------------------------------------

force = function(x,y){
  -16*y + 15*exp(-x)
}


# Compute Algorithm -------------------------------------------------------

for(i in 0:(L/h)){

  k1 = h*force(x,y[i])
  k2 = h*force(x + h/2, y[i] + k1 /2)
  k3 = h*force(x + h/2, y[i] + k2 /2)
  k4 = h*force(x + h  , y[i] + k3   )

  temp=y[i] + (1/6)*(k1 + 2*k2 + 2*k3 + k4)

  y = c(y,temp)
  x = x + h
  i = i+1  
}

t <- seq(from=0, to=L, by=h)
plot(t,y,type="l")

MATLAB Script:

%% Initialise Variables
y0 = 0;
x0 = 0;
L  = 1; %Length of Domain of function (here y)
N  = 100; %Number of steps to break the domain into
h  = L/N; %Step Size

yi = y0; %intermediate value of y
xi = x0; %intermediate value of x to be ticked in algo

y = zeros(N,1); %store y values as components
x = 0:h:L; %just for plot later

%% Define Forcing Function
syms f(A,B)
f(A,B) = 15*exp(-A) - 16*B;

%% Execute Algorithm
for n = 1:1:N;
    xi= h*(n-1);

    k1= h*f(xi,yi);    
    k2= h*f(xi + h/2 , yi + k1/2);
    k3= h*f(xi + h/2 , yi + k2/2);
    k4= h*f(xi + h   , yi + k3  );

    yi= yi + (1/6)*(k1 + 2*k2 + 2*k3 + k4);
    y(n,1)=yi;
end

%plot(x,y)

Answer 1

The reason you have this problem is because you're using symbolic variables to do a computation that should be purely numerical.

If you define f as follows:

f = @(A,B)15*exp(-A) - 16*B;

The loop finishes almost instantly. Few more notes:

• The syntax above is used to define a function handle to an anonymous function.
• The resulting x and y vectors have a different length, so you wouldn't be able to plot them afterwards.
• In the future you should profile your code to find performance bottlenecks.

P.S.
The MATLAB equivalent of the function definition you had in R would be quite similar:

function out = force(x)
  out = 15*exp( -x(1) ) - 16*x(2);
end

or

function out = force(x,y)
  out = 15*exp(-x) - 16*y;
end

... depending on whether the input comes as a vector or not.